DESIGN OF WHITWORTH QUICK RETURN MECHANISM USING … · Fig 1.2: Velocity profile of slider of...

International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 2, Issue- 6, June-2014

Design of Whitworth Quick Return Mechanism Using Non-Circular Gears

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DESIGN OF WHITWORTH QUICK RETURN MECHANISM USING NON-CIRCULAR GEARS

1JASTI SAI PRANEETH, 2ANIL KUMAR SREEDHARALA, 3NAVEEN KUMAR DARISI,

4M ELANGOVAN

1Student, 2Student, 3Student, 4Professor E-mail: [email protected], [email protected], [email protected],

[email protected]

Abstract- In this paper, a mechanism similar in function to Whitworth quick return mechanism is designed using a pair of non-circular gears integrated into Scotch-yoke mechanism. The velocity profile of the slider in quick return mechanism is derived analytically and used as reference to the velocity of slider in proposed design. Keywords-Elliptical gears, non-circular gears, Scotch-yoke mechanism, Whitworth quick return mechanism I. INTRODUCTION Circular gears are widely used in design of wide range of mechanisms. There are no practical limitations to the usage of gears but their versatilities can be maximised when used as non-circular gears. Non-Circular gears are gears with distorted shapes that are available mostly as elliptical gears, eccentric circular gears, oval gears, or internal noncircular gears and of higher order. Non-Circular gears are used to improve the function, versatility, and simplicity of many mechanical processes. Noncircular gears have been considered a curiosity and a product of niche applications for a long time because of their design and manufacture complexity. Our paper tries to contemplate a better inside by designing an application based on non-circular gears. Whitworth Quick-Return Mechanism is second inversion of the slider crank mechanism attained by fixing the crank instead of the ground link. The peculiar feature of this mechanism is that its forward stroke takes a little longer and cuts the metal whereas the return stroke takes a shorter period. The shaper machine used in lathe workshops works on the principle of Whitworth quick return mechanism (as shown in Fig 1.1). Quick return mechanism consists of many linkages that rotates at high speeds. The linkages tend to have dynamic effects. Balancing weight are to be added to these linkages to avoid vibrations. This intern increases the overall weight of the mechanism. The proposed mechanism intended to replace the conventional mechanism with a compact, robust design.

Fig1.1: Whitworth Quick Return Mechanism

The velocity profile of the slider in quick return mechanism is shown in Fig 1.2. This paper tries to recreate the features of the profile.

Fig 1.2: Velocity profile of slider of Whitworth quick return mechanism using MATLAB The scotch yoke mechanism is a very simple mechanism for converting rotary motion into reciprocating motion. This mechanism is basically a second inversion of double slider crank mechanism



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obtained when any of the sliding blocks is fixed. The velocity profile of the slider in scotch-yoke mechanism is sinusoidal in nature as shown in Fig 1.3. This profile is to be imposed by another function that is

created by a pair on non-circular gears inorder to create a velocity profile similar to that of quick return mechanism as shown in Fig 1.2

Fig 1.3: Velocity profile of slider in Scotch-yoke mechanism

II. MODIFIED SCOTCH-YOKE MECHANISM:

The Scotch–Yoke mechanism in the proposed design is driven by a pair of non-circular gears 1 and 2 and two circular gears 3 and 4 (Fig. 2.1(a)) so that the

input to the conventional scotch-yoke mechanism is no longer a constant i.e., ≠ constant (Fig 2.1(b)).

Fig 9.2.1: Composition of gear drive with scotch-yoke mechanism: (a) eccentric gear drive and scotch-yoke mechanism at their initial

positions and (b) links of scotch yoke mechanism and details of related velocities



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The purpose of this design is to improve the function of output velocity of the Scotch-Yoke mechanism (coupled with gears) with the following conditions.

1. Gear 1 (Fig. 2.1(a)) is an elliptical involute gear and noncircular gear 2 is conjugated to gear 1.

2. Gears 3 and 4 (Fig. 2.1(a)) are circular gears

with gear ratio of 1 ( ); gear 4

performs one revolution for one revolutions of gear 3.

3. Function is of a period φ1 = 2π, and this is obtained by application of circular gears 3 and 4 in addition to elliptical gear 1 and noncircular gear 2.

The basic kinematic relation of the coupled mechanism is based on the drawings of Figure 2.1(a) shows that the initial position of links at φ1 = 0.

The basic kinematic relations of the velocities of the links of the mechanism are represented by the following equations (2.1) Whereas is the angle of rotation of the crank. Such a relation is a sine type function. Angular velocity is determined as

= = = =

(2.2) Angle of rotation is also derived as

= = = =

[ ] (2.4) Fig.9.2.2 shows the velocity function for different values of parameter of eccentricity e1 when a tandem design based on one-pair of elliptical gears and a Scotch–Yoke mechanism is applied.

Fig 9.2.2: Velocity of the scotch–yoke as a function of angle of rotation of an elliptical gear for different values of the parameter of

eccentricity e1

III. DESIGN OF ELLIPTICAL GEARS In figure 3.1, the noncircular gears (elliptical gears) 1 and 2 with angular velocity ω1 and ω2 and angular

displacement φ1 and φ2, respectively, reversely rotate around the fixed axes O1 and O2. Coordinates systems X1Y1 and X2Y2 are the moving coordinate attached to gears 1 and 2, respectively.

Fig 3.1: Illustration of two initial positions of gear 1 and 2 where O1 is (a) the lower focus of ellipse 1 ; (b) the upper focus of ellipse 1



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The driving centrode is represented in polar form by

r1 ( ), 0 ≤ φ1 ≤ 2π n (3.1) The rotating- speed ratio or transmission ratio m21 (φ1) is:

m21 ( ) = = = (3.2)

Where E = + is the center distance. Similarly, derivative function m21 (φ1) is represented by

m12 ( ) = = = (3.3)

The meaning of Eqn. (3.2) and (3.3) is that centrodes σ1 and σ2 roll over each other. Function m12 ( ) must be a periodic one to obtain a closed form curve

m21 (φ1) = = = (3.4)

When =0 then also be zero. Integrating equation (3.3.1) with respect to φ1, the relationship of φ1 and φ2 is obtained:

= = =

(3.5)

Where n is an integer number and represents the number of revolutions of the driving centrode for one revolution of the driven centrode.

Eqn. 3.3 can be interchanged so that the centrode of elliptical gears can be defined as

r1 (φ1) = & (3.6)

r2 (φ2) = . (3.7)

A conventional elliptical gear drive is formed by two elliptical gears that may be assembled as shown in Figs. 3.1(a) or 3.1(b). Figure 3.1(a) shows a gear drive wherein centre O1 of rotation of centrode σ1 coincides with the lower focus F of ellipse 1. On the other side, Figure 3.1(b) shows a gear drive wherein centre O1 of rotation of centrode σ1 coincides with the upper focus F∗ of ellipse 1. Figure 3.2 shows an elliptical centrode that performs rotation around centre O1 (it is the lower focus F of ellipse 1). We recall that the instantaneous point I of tangency of centrode 1 and centrode 2 (centrode 2 is not shown) moves along centre distance O1O2 (Fig. 3.2) in the process of motion. The centre distance is E = 2a, where 2a is the major axis of mating elliptical centrodes 1 and 2. Centrode σ1 (Figs. 3.1(a) and 3.2) is represented by

r1( 1) = = (3.8)

Fig 3.2: For derivation of transmission function of elliptical gears

The derivative function m12 (Φ1) is derived to

m12(Φ1) = = = (3.9)

where determines the polar equation of centrode. Taking into account Eq.(3.8), E = 2a, and p = a(1 − e2), we obtain,

m12(Φ1) = (3.10)

The sought-for transmission function may then be represented as

Φ2(Φ1) = =

(1-e2) (3.11)

We obtain transmission function Φ2(Φ1) as a closed form solution as follows

tan = tan (3.12)

The transmission function represented in Eq. (3.12) corresponds to the assembly positions of centrodes as shown in Fig. 3.1(a). For assembly as shown in Fig. 3.1(b), we obtain



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tan = tan (3.13)

IV. PROPOSED DESIGN From the Eqn. 2.2 we derived the angular velocity as

= = = = (2.2)

Wherein is derived earlier as

(4.1)

Hence, =

(4.2) Where e is the eccentricity of the ellipse. Angle of rotation is also derived as

= = = =

[ ] (4.3) Eqn. (2.1) – (7.2.4) yield

= Sin( [ ] ) (4.4) Eqs. (2.1), (4.2), (4.3), (4.4) yield

= Sin( [ ] ) (4.5) A matlab code was generated to design the pair of elliptical gears based on the mathematical equations derived earlier. Refer Appendix-I for code to generate centrode of elliptical gears. The Cartesian coordinates of these plots are imported into Solidworks through the module, curve through XYZ. Once they were imported into Solidworks, tooth profile needs to be generated over the centrodes. For this a circle with radius Ρmin is drawn tangent to the centrode as shown in Fig 3.3

Where, Ρmin = = a(1-e2) (4.6)

The number of teeth of an elliptical gear is determined by

N = (4.7)

Once the tooth is drawn it is arrayed on the surface not along the circle but along the curve imported earlier. On completion of tooth profile the elliptical gear design is completed. The other conjugate elliptical gear is also designed in the same procedure.

Fig3.3:Tooth profile generated onto the centrode of elliptical

gear A prototype model of Scotch-yoke mechanism integrated with noncircular elliptical gears is designed as shown in Fig. 3.4

CONCLUSION The velocity profile of the slider is plotted under given conditions and parameters using MATLAB as shown in Fig. 11.1

Fig 11.1: Velocity profile of slider of Scotch-yoke mechanism

coupled with elliptical gears



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On careful inspection of the profile it is evident that while return stroke i.e., when crank angular displacement increases from 00 to 900 , the velocity of slider increase gradually and then dwells at the maximum velocity during entire return stroke. During forward stroke the velocity of slider drops gradually to zero by the time the slider is at full stroke. This evidently proves that the elliptical gears are designed correctly and an alternate mechanism to that of quick return mechanism is designed successfully. REFERENCES

[1]. Bair, B.W., 2002, “Computer Aided Design of Elliptical Gears,” Transactions of the ASME, Journal of Mechanical Design, Vol.124, pp.787-793.

[2]. Bair, B.W., 2004, “Computer Aided Design of Elliptical Gears with Circular-arc Teeth,” Mechanism and Machine Theory, Vol.39, No.2, pp.153-168.

[3]. Bair, B W., “Computer aided design of non-standard elliptical gear drives,” Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 2002, 216(4):473-483.

[4]. Danieli, G.A., 2000, “Analytical Description of Meshing of Constant Pressure Angle Teeth Profile on Variable Radius Gear and its Applications” Transactions of the ASME, Journal of Mechanical Design, Vol.122, pp.123-129.

[5]. Faydor L. Litvin, Ignacio Gonzalez-Perez, Alfonso Fuentes, Kenichi Hayasaka,” Noncircular Gears: Design and Generation,” Cambridge university press, United states of America, 2009

[6]. G.A. Danieli, D. Mundo, “New developments in variable radius gears using constant pressure angle teeth profiles,” Mechanism and Machine Theory 40 (2005) 203–217.

[7]. Liu, Jen-Yu and Chen, Yen-Chuan, “A design for the pitch curve of noncircular gears with function generation”, “Proceedings of the International Multi Conference of Engineers and Computer Scientists”, Vol II, 2008.

[8]. Ottaviano, Erika, et al., ”Numerical and experimental analysis of non-circular gears and cam-follower systems as function generators”, “Mechanism and machine theory”, Vol. 43, 996-1008, 2008.

[9]. S.L. Chang, C.B. Tsay, “Computerized tooth profile generation and undercut analysis of noncircular gears manufactured with shaper cutters,” Journal of Mechanical Design, ASME Transactions 120 (1998) 92–99.

[10]. X. Wu, S. Wang, A. Yang, “Non-circular gear CAD/CAM technology, in: Proceedings of the Eighth World Congress on the Theory of Machines and Mechanisms, Prague, Czechoslovakia, 1991, pp. 391–394.

[11]. Yongping Liu, Pengfei Meng,” Parametric design of the pitch curves of higher-order elliptic gears,” Science Technology and Engineering, 2010, 10(1): 39-42.

[12]. Vasie Marius, Andrei Laurenţia,” Technologies for Non-Circular Gear Generation and Manufacture” , The Annals “Dunărea De Jos” Of Galaţi Fascicle V, Techolologies in Machine Building, ISSN 1221-4566, 2010.

APPEDIX I MATLB CODE FOR GENERATION OF CENTRODE OF A PAIR OF ELLIPTICAL GEARS % assigning variables % >> e = .1; % transmission ratio of circular gears% >> y = 1; % Centre distance between elliptical gears% >> a = 50; >> x = (0 : sym ('1/360'): 1); >> theta = 2*pi*x; >> for n = 1:361 % defining the transmission ratio of elliptical gears% m(n) = (1+(e^2)-2*e.*cos(theta(n)))/(1-e^2); % Defining the centrodes of both gears% r1(n) = a*(1-e^2)/(1-e*cos(theta(n))); r2(n) = 2*a - r1(n); %Deriving the phase change between the gears% theta2(n) = theta(n) + 2*atan((11*tan(theta(n)/2))/9) - 2*atan(tan(theta(n)/2)); angle(n) = atan(((1+e)/(1-e)*tan(theta(n)/2))); end % Rectifying the errors of phase% >> for n = 1:361 if n>=1 & n <=181 sample(n) = sin(y*angle(n)); end if n>181 sample(n) = -sin(y*angle(n)); end end % plotting the centrodes of both gears% >> plot(r1.*sin(theta),r1.*cos(theta)) >> plot(r2.*sin(theta),r2.*cos(theta)) %Plotting the velocity profile of output slider% >> plot(theta,sample./m)

DESIGN OF WHITWORTH QUICK RETURN MECHANISM USING … · Fig 1.2: Velocity profile of slider of...

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